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We investigate the medium modifications of the masses of pseudoscalar open
charm ($D$ and $\bar D$) mesons and the charmonium state ($\psi(3770)$) in hot
isospin asymmetric strange hadronic medium in the presence of an external
magnetic field. The partial decay width of $\psi(3770)$ to $D\bar D$, arising
due to the mass modifications of these mesons, in the magnetized matter, is
studied using a light quark pair creation model, namely the $^3P_0$ model. The
in-medium properties are studied incorporating the effects due to the baryonic
Dirac sea as well as the effects due to mixing of the pseudoscalar (S=0) and
vector (S=1) mesons (PV mixing). The mass modifications of the open charm
mesons are computed from their interactions with the baryons (nucleons and
hyperons) and the scalar mesons and that of the charmonium state is calculated
from the medium change of scalar dilaton field, which mimics the gluon
condensates within a chiral effective model. For magnetized nuclear matter, at
$\rho_B=\rho_0$, when the nucleon anomalous magnetic moments (AMMs) are taken
into account, the effect from Dirac sea gives rise to inverse magnetic
catalysis (MC), which is decrease of the magnitude of the scalar field
(proportional to the light quark condensates) with increase in the magnetic
field, whereas, the opposite effect is observed when the AMMs are ignored.
However, the inclusion of hyperons in the medium is observed to lead to a
magnetic catalysis, when the AMMs of the baryons are taken into account. The
present study of the charm mesons in the magnetized isospin asymmetric strange
hadronic matter at finite temperatures can have observable consequences on the
production of these mesons in ultra-relativistic peripheral heavy ion collision
experiments.